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Chapter 9: Problem 63

Solve using the quadratic formula. $$ t 2-5 t+10=0 $$

Short Answer

Expert verified
The equation has two complex roots: \( t = \frac{5 \pm \sqrt{15}i}{2} \).

Step by step solution

01

Identify the coefficients

The general form of a quadratic equation is \( at^2 + bt + c = 0 \). Here, identify the coefficients: \( a = 1 \), \( b = -5 \), and \( c = 10 \).
02

Write the quadratic formula

The quadratic formula used to find the roots of a quadratic equation is \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
03

Calculate the discriminant

The discriminant is the expression under the square root in the quadratic formula: \( b^2 - 4ac \). Substitute the values: \( (-5)^2 - 4 \cdot 1 \cdot 10 = 25 - 40 = -15 \).
04

Determine nature of roots

Since the discriminant is negative (\(-15\)), the quadratic equation has two complex roots.
05

Compute the complex roots

Substitute the coefficients and the discriminant into the quadratic formula: \( t = \frac{-(-5) \pm \sqrt{-15}}{2 \cdot 1} \). Simplify to find the roots: \( t = \frac{5 \pm \sqrt{15}i}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equation
A quadratic equation is a polynomial equation of degree two, which can be written in the general form \( ax^2 + bx + c = 0 \). In this equation:
  • \( a \), \( b \), and \( c \) are coefficients, with \( a eq 0 \).
  • \( x \) is the variable or the unknown number we want to solve for.
  • The term \( ax^2 \) is known as the quadratic term, \( bx \) is the linear term, and \( c \) is the constant term.
To solve a quadratic equation like \( t^2 - 5t + 10 = 0 \), where \( a=1 \), \( b=-5 \), and \( c=10 \), we often use the quadratic formula. This method covers all types of roots, including real and complex roots, and is especially helpful for equations that cannot be easily factored.
Complex Roots
Roots of a quadratic equation are values of \( x \) that satisfy the equation. In some cases, these roots can be complex numbers. A complex number has a real part and an imaginary part, often written in the form \( a + bi \), where \( i \) is the imaginary unit satisfying \( i^2 = -1 \).
When using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), if the value under the square root (the discriminant) is negative, the square root won't be a real number, leading to complex roots.
  • Example: In \( t = \frac{5 \pm \sqrt{15}i}{2} \), the roots are complex because of the imaginary unit \( \sqrt{-15} = \sqrt{15}i \).
  • These roots are expressed in a pair of complex numbers, where one root is the complex conjugate of the other.
Discriminant
The discriminant is a key part of the quadratic formula, calculated as \( b^2 - 4ac \). It determines the nature of the roots of a quadratic equation:
  • If the discriminant is positive, the equation has two distinct real roots.
  • If it is zero, there is one real root (or a repeated real root).
  • If it is negative, the equation has two complex roots, as seen in our example where \( b^2 - 4 \cdot 1 \cdot 10 = -15 \).
Understanding the discriminant helps predict whether the solutions will involve real numbers or include imaginary components, which is crucial for solving equations correctly.

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